In this paper we prove that the Landau-Kolmogorov inequality for functions on the half line holds for Lorentz spaces with the constants, which are best possible for $L_\infty$-space.
"A Landau-Kolmogorov Inequality for Lorentz Spaces." Tokyo J. Math. 27 (1) 13 - 20, June 2004. https://doi.org/10.3836/tjm/1244208470